Nonlinear eigenvalue problems for coupled Helmholtz equations modeling gradient-index graphene waveguides

Jung Heon Song, Matthias Maier, Mitchell Luskin

Research output: Contribution to journalArticlepeer-review

Abstract

We discuss a quartic eigenvalue problem arising in the context of an optical waveguiding problem involving atomically thick 2D materials. The waveguide configuration we consider consists of a gradient-index (spatially dependent) dielectric equipped with conducting interior interfaces. This leads to a quartic eigenvalue problem with mixed transverse electric and transverse magnetic modes, and strongly coupled electric and magnetic fields. We derive a weak formulation of the quartic eigenvalue problem and introduce a numerical solver based on a quadratification approach in which the quartic eigenvalue problem is transformed to a spectrally equivalent companion problem. We verify our numerical framework against analytical solutions for prototypical geometries. As a practical example, we demonstrate how an improved quality factor (defined by the ratio of the real and the imaginary part of the computed eigenvalues) can be obtained for a family of gradient-index host materials with internal conducting interfaces. We outline how this result lays the groundwork for solving related shape optimization problems.

Original languageEnglish (US)
Article number109871
JournalJournal of Computational Physics
Volume423
DOIs
StatePublished - Dec 15 2020

Bibliographical note

Funding Information:
The authors were supported in part by ARO MURI Award W911NF-14-1-0247.The first and third author's research was supported in part by NSF Awards DMS-1819220 and DMS-1906129.This author's research was supported in part by NSF Award DMS-1912847.

Publisher Copyright:
© 2020 Elsevier Inc.

Keywords

  • Guided mode
  • Nonlinear eigenvalue problem
  • Quadratification
  • Quartic eigenvalue problem
  • Surface plasmon polariton
  • Time-harmonic Maxwell's equations

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